Exercise 111.12.7. Let $(R, \mathfrak m)$ be a Noetherian local ring. Assume that $R$ is reduced, i.e., $R$ has no nonzero nilpotent elements. Assume moreover that $R$ has two distinct minimal primes $\mathfrak p$ and $\mathfrak q$.

Show that the sequence of $R$-modules

\[ 0 \to R \to R/\mathfrak p \oplus R/\mathfrak q \to R/\mathfrak p + \mathfrak q \to 0 \]is exact (check at all the spots). The maps are $x \mapsto (x \bmod \mathfrak p, x \bmod \mathfrak q)$ and $(y \bmod \mathfrak p, z \bmod \mathfrak q) \mapsto (y - z \bmod \mathfrak p + \mathfrak q)$.

Show that if $\text{depth}(R) \geq 2$, then $\dim (R/\mathfrak p + \mathfrak q) \geq 1$.

Show that if $\text{depth}(R) \geq 2$, then $U = \mathop{\mathrm{Spec}}(R) \setminus \{ \mathfrak m\} $ is a connected topological space.

This proves a very special case of Hartshorne's connectedness theorem which says that the punctured spectrum $U$ of a local Noetherian ring of $\text{depth} \geq 2$ is connected.

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